Let us start by writing the integral equation for the vector potential where we presume that we've already transformed the IWE into the IHE. We will choose to use the outgoing wave Green's function to make it clear that the field we are looking for is the one that the source is emitting, not one that it is absorbing.

(11.68) |

Note that a more general solution would be one that allowed for absorption of incoming waves as well as the emission of outgoing waves, but that this would require knowing something about the sources outside the domain considered to be infinite. We will talk about this later (scattering theory and the optical theorem).

From
we can easily find
or
:

(11.69) |

(11.70) |

(11.71) | |||

(11.72) |

or

(11.73) |

Doing the integral above can be quite difficult in the general case.
However, we'll find that for most *reasonable, physical situations*
we will be able to employ certain approximations that will enable us to
obtain a systematic hierarchy of descriptions that converge to the
correct answer as accurately as you like, at the same time they increase
our physical insight into the radiative processes.